JEE Main & Advanced Mathematics Geometric Progression Geometric Mean

Geometric Mean

Category : JEE Main & Advanced

If \[a,G,b\] are in G.P., then G is called G.M. between \[a\] and \[b\].

 

(1) If \[a,\,{{G}_{1}},\,{{G}_{2}},\,{{G}_{3}},....\,{{G}_{n}},\,b\] are in G.P. then \[{{G}_{1}},\,{{G}_{2}},\,{{G}_{3}},....\,{{G}_{n}}\] are called n G.M.’s between \[a\] and \[b\].

 

(2) Insertion of geometric means : (i) Single G.M. between a and b : If a and b are two real numbers then single G.M. between \[a\] and \[b\]\[=\sqrt{ab}\].

 

(ii) n G.M.’s between a and b : If \[{{G}_{1}},\,{{G}_{2}},\,{{G}_{3}},\,......,\,{{G}_{n}}\] are n G.M.’s between a and b, then

 

\[{{G}_{1}}=ar=a{{\left( \frac{b}{a} \right)}^{\frac{1}{n+1}}}\], \[{{G}_{2}}=a{{r}^{2}}=a{{\left( \frac{b}{a} \right)}^{\frac{2}{n+1}}}\],

 

\[{{G}_{3}}=a{{r}^{3}}=a{{\left( \frac{b}{a} \right)}^{\frac{3}{n+1}}}\], ……………….., \[{{G}_{n}}=a{{r}^{n}}=a{{\left( \frac{b}{a} \right)}^{\frac{n}{n+1}}}\].



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